How do you find the domain of log(x^2-4)?
and how about this one? e^(3x+4)?
Please help me!
Well kristie, you might recognize log(x^2-4) = log((x-2)(x+2)) and then use a property of logs to see it as a sum, i.e., log(ab)=log(a)+log(b). You are then expected to know that the log function is only defined for positve values. Thus you only want x values that make the argument positive.
For the second you want the values that e^y is defined, where y = 3x+4. What values can an exponent be?
thanks, Roger!
I get the first one, but I'm still a little confused about the second one. I thought an exponent could be any values. So would that make (-infinity, _infinity)?
yes, I think you understand it. Exponents can assume any values.
Just beware of a number with a fractional exponent, such as e^(1/(x-1)) for example. Here you need to restrict x not equal 1.
That's correct, Roger! Exponents can indeed assume any real values, which means that the domain of the function e^(3x+4) is (-infinity, infinity), as you mentioned. However, you brought up a good point about fractional exponents.
When dealing with fractional exponents, we have to be careful because certain values may not be defined. For example, in the function e^(1/(x-1)), the exponent has a fraction with a denominator of (x-1). In this case, the exponent is defined as long as (x-1) is not equal to zero. So we need to exclude the value x = 1 from the domain of the function.
In summary, most exponential functions have a domain of (-infinity, infinity), but when dealing with fractional exponents or expressions in the exponent, we need to consider any restrictions or exclusions that might arise.